Collapse of random triangular groups: a closer look

Abstract

The random triangular group \Gamma(n,t) is a group given by a presentation P= , where S is a set of n generators and R is a random set of t cyclically reduced words of length three. The asymptotic behavior of \Gamma(n,t) is in some respects similar to that of widely studied density random group introduced by Gromov. In particular, it is known that if t <= n^{3/2-\epsilon} for some \epsilon > 0, then with probability 1-o(1) \Gamma(n,t) is infinite and hyperbolic, while for t >= n^{3/2+\epsilon}, with probability 1-o(1) it is trivial. In this note we show that \Gamma(n,t) collapses provided only that t <= C n^{3/2} for some constant C>0.